408 XV NORMED ALGEBRAS AND SPECTRAL THEORY 16. Let X be a locally compact space, p a bounded positive measure on X with total mass 1, and u a /z-measurable mapping of X into X such that /x is invariant with respect to u (Section 13.9, Problem 24). Let U be the unitary operator on L£(X, /x) such that £/./=(/oW)~ (Section 13.11, Problem 10). (a) The mapping u is said to be mixing (resp. weakly mixing ) with respect to /LI if, for each pair of /^-measurable subsets A, B of X we have lim (M«-fl(A) n B)) = /i(A)/*(B) 11-* 00 (resp. lim - "l |/*(ir *CA) n B) - /*(A)/«(B)| = 0.) n-*o> n ks;Q Every mixing mapping is weakly mixing. Every weakly mixing mapping is ergodic (Section 13.9, Problem 13(d)). (b) Show that u is mixing (resp. weakly mixing) if and only if, for each pair of functions/, g in ^c(^» A0» we nave (resp. An equivalent condition is that, for each / e .#c(X» p) such that (/11) = 0 (i.e., If dp, = 0), we have lim 0/M-/I/) = 0 n-»oo (resp. lim I "SKI/" -f\S) - (/I D(l \9) I = 0). (Replace / by /+ g, and remark that if a sequence (<?„) satisfies the condition •1 »~1 1 n~1 lim - ^ \ak\2 = 0, then also lim - ]£ \ak\ = 0, by the Cauchy-Schwarz inequality.) n-*oo n IcssO «-*« n k*=Q (c) For u to be ergodic with respect to /x, it is necessary and sufficient that 1 should be an eigenvalue of U with multiplicity 1. If this is so, then all the eigen- values of U have multiplicity 1 and form a subgroup of the group U of complex numbers of absolute value 1. For each eigenvector /e Lc(X, (JL) of £/, the function |/| is constant almost everywhere. (Remark that if £/-/=A/and U - g = \g, then (d) Show that the following properties are equivalent : (a) u is weakly mixing with respect to /it. (/?) u x u is an ergodic mapping of X x X into X x X with respect to the measure /u, (g) /x. (y) The only eigenvalue of U is 1. (To show that (a) implies (/?), consider subsets M x N of X x X, where M and N are /^-measurable subsets of X. To prove that (/8) implies (y), observe that if /is and K a compact